Face numbers of sequentially Cohen–Macaulay complexes and Betti numbers of componentwise linear ideals

Abstract

A numerical characterization is given of the $h$-triangles of sequentially Cohen–Macaulay simplicial complexes. This result determines the number of faces of various dimensions and codimensions that are possible in such a complex, generalizing the classical Macaulay–Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree $\le d$ and shifted pure $(d–1)$-dimensional simplicial complexes